Parallel Triangle Counting in MPI Jason Li and David Wise - - PowerPoint PPT Presentation

Parallel Triangle Counting in MPI

Jason Li and David Wise

Background

• A triangle in a undirected

graph is a collection of 3 vertices such that all 3 pairs of vertices are connected by an edge.

• “Triangle counting has

emerged as an important building block in the study of social networks, identifying thematic structures of networks, spam and fraud detection, link classification and recommendation, and more” [1]

Background

• A triangle in a undirected

graph is a collection of 3 vertices such that all 3 pairs of vertices are connected by an edge.

• “Triangle counting has

emerged as an important building block in the study of social networks, identifying thematic structures of networks, spam and fraud detection, link classification and recommendation, and more” [1]

This graph has 2 triangles:

The Underlying Algorithm

• Initialize the counter to 0.
• Sort the vertices in order of increasing degree, breaking ties arbitrarily.

Similarly, sort the adjacency lists according to the same ordering.

• For each edge (v, w) with v < w:
• Let uv and uw be the first vertices in the adjacency lists of v and w

respectively.

• While uv exists and uv < v and uw exists and uw < v:
• If uv < uw then set uv to the next neighbor of v.
• Else if uw < uv then set uw to the next neighbor of w.
• Else increment the counter and set uv and uw to their next neighbors.

Complexity of the Algorithm

• The space complexity is just O(m) since we

store the graph

• Because the vertices are sorted by degree and

each edge is assigned to its smaller neighbor, it can be shown that the sequential time complexity is O(m3/2).

Parallelizing the Algorithm

• The focus of our project was efficiently parallelizing

this algorithm

• Naive idea: each edge is a task and can be arbitrarily

assigned to a processor

• The catch is that to process an edge, the processor

needs to know the neighbors of each vertex on the edge

• If the edges are arbitrarily assigned, each

processor needs a copy of the whole graph

Reducing Communication

• We want the edges assigned to each processor to hit

as few vertices as possible

• We can approach the problem by grouping the

vertices

• We partition the vertices into r = √P groups


v1, …, vr and assign each processor a pair
 (vi, vj)

• The processor assigned pair (vi, vj) is responsible

for all edges going from a vertex in vi to vj.

Cost Analysis

• In the average case,

each processor gets 1/P edges: we expect near perfect speedup

• Each processor gets

the adjacency lists

• f 2n/√P vertices,

which on average has total size 2m/√P

4 processors 2 groups of vertices P1 P2 P3 P4

The thick arrows represent groups of edges.

Actual Speedups Matched Expectations

Time (s) Number of Processors 1 4 9 16 25 36 49 64

More Results

gplus k5000 live skitter Speedup 17.23 12.72 1.95 3.98

Thank you!

Questions?

References

• 1. “Counting and Sampling Triangles from a Graph Stream” A. Pavan, Kanat

Tangwongsan, Srikanta Tirthapura, Kun-Lung Wu, Proceedings of the VLDB Endowment VLDB Endowment Hompage archive, Volume 6 Issue 14, September 201, Pages 1870-1881.

• 2. “15-418 Final Report”, Shu-Hao Yu, YiCheng Qin. http://www.cs.cmu.edu/afs/

cs/user/shuhaoy/www/Final_Project.pdf.